Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.107·3-s + 0.451·5-s + 2.31·7-s − 2.98·9-s + 2.84·11-s + 2.31·13-s − 0.0485·15-s + 0.174·17-s − 2.11·19-s − 0.248·21-s + 0.730·23-s − 4.79·25-s + 0.642·27-s − 7.10·29-s − 10.1·31-s − 0.305·33-s + 1.04·35-s − 2.63·37-s − 0.248·39-s − 7.51·41-s − 0.351·43-s − 1.35·45-s − 6.18·47-s − 1.63·49-s − 0.0187·51-s − 0.440·53-s + 1.28·55-s + ⋯
L(s)  = 1  − 0.0619·3-s + 0.202·5-s + 0.875·7-s − 0.996·9-s + 0.857·11-s + 0.641·13-s − 0.0125·15-s + 0.0424·17-s − 0.484·19-s − 0.0542·21-s + 0.152·23-s − 0.959·25-s + 0.123·27-s − 1.31·29-s − 1.81·31-s − 0.0531·33-s + 0.176·35-s − 0.432·37-s − 0.0397·39-s − 1.17·41-s − 0.0535·43-s − 0.201·45-s − 0.901·47-s − 0.232·49-s − 0.00262·51-s − 0.0605·53-s + 0.173·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6008\)    =    \(2^{3} \cdot 751\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;751\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;751\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 + 0.107T + 3T^{2} \)
5 \( 1 - 0.451T + 5T^{2} \)
7 \( 1 - 2.31T + 7T^{2} \)
11 \( 1 - 2.84T + 11T^{2} \)
13 \( 1 - 2.31T + 13T^{2} \)
17 \( 1 - 0.174T + 17T^{2} \)
19 \( 1 + 2.11T + 19T^{2} \)
23 \( 1 - 0.730T + 23T^{2} \)
29 \( 1 + 7.10T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 2.63T + 37T^{2} \)
41 \( 1 + 7.51T + 41T^{2} \)
43 \( 1 + 0.351T + 43T^{2} \)
47 \( 1 + 6.18T + 47T^{2} \)
53 \( 1 + 0.440T + 53T^{2} \)
59 \( 1 + 1.57T + 59T^{2} \)
61 \( 1 + 4.03T + 61T^{2} \)
67 \( 1 + 4.68T + 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 3.78T + 73T^{2} \)
79 \( 1 - 16.4T + 79T^{2} \)
83 \( 1 + 7.15T + 83T^{2} \)
89 \( 1 - 3.87T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.914422450998091835442990077705, −6.96003160392955576650800905175, −6.25308687301961274689345013913, −5.55067627032629291119821818081, −4.99791215522874238482907332487, −3.89703604187509157837270480876, −3.41444921120647940298356414113, −2.09509530110269215155165051449, −1.51628467075664754916822197814, 0, 1.51628467075664754916822197814, 2.09509530110269215155165051449, 3.41444921120647940298356414113, 3.89703604187509157837270480876, 4.99791215522874238482907332487, 5.55067627032629291119821818081, 6.25308687301961274689345013913, 6.96003160392955576650800905175, 7.914422450998091835442990077705

Graph of the $Z$-function along the critical line